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Chapter 7b 7b-1 Business Statistics: A First Course 5th Edition Learning Objectives In this part of the chapter, you learn: To compute probabilities from the normal approximation of the binomial distribution ! To determine probabilities from the logarithmic distribution ! Chapter 7 Part b Continuous Probability Distributions BUS210: Business Statistics Continuous Probability Distributions - 2 Normal Approximation Probability Distributions Probability Distributions Discrete Probability Distributions Binomial Normal Poisson Exponential ! Continuous Probability Distributions - 3 ! In this case, the normal probability distribution provides an easy-to-use approximation of binomial probabilities BUS210: Business Statistics Continuous Probability Distributions - 4 Normal Approximation of the Binomial of the Binomial Example: Coffee and a Donut Recall from the binomial that! and ! = n" (1! " ) ! A large donut company states that 60% of their customers order coffee along with their donut. What is the probability that 11 of next 15 customers will order coffee? We can use that information in the formula z= ! A dataset is large whenever the following are all true: ! n > 20, and ! n! > 5, and Note: " Some textbooks use " n! > 10 and n(1-!) > 10" ! n(1 - !) > 5. Normal Approximation µ = n! ! ! Ch. 7 BUS210: Business Statistics When the number of trials, n, becomes large, evaluating the binomial probability function by hand or even with a calculator becomes difficult. ! Continuous Probability Distributions Ch. 6 of the Binomial x!µ ! ! First, calculate mean and standard deviation: µ = n! = 15(0.6) = 9 However, we must add or subtract a 0.5 continuity correction factor to x: • • ! = n" (1# " ) = 9(0.4) = 1.897 A discrete distribution is approximated by a continuous. Example: P(x = 10) is approximated by P(9.5 < x < 10.5). Note: The correction factor should always EXPAND the area of interest. BUS210: Business Statistics BUS210 Business Statistics Continuous Probability Distributions - 5 BUS210: Business Statistics Continuous Probability Distributions - 6 NSCC Chapter 7b 7b-2 Normal Approximation of the Binomial Probability Distributions Probability Distributions Example: Coffee and a Donut (Continued) ! Next, restate the problem in terms of the correction factor: Discrete Probability Distributions P(10.5 ! x ! 11.5) ! Next, solve for area under the normal curve: zL = x L ! µ 10.5 ! 9.0 = = 0.79 " 1.897 zU = xU ! µ 11.5 ! 9.0 = = 1.32 " 1.897 P(10.5 ! x ! 11.5) = P(0.79 ! z ! 1.32) = P(z ! 1.32) " P(z ! 0.79) = 0.4066 " 0.2852 = 0.1214 Continuous Probability Distributions Binomial Normal Poisson Exponential Ch. 6 BUS210: Business Statistics ! ! Ch. 7 BUS210: Business Statistics Continuous Probability Distributions - 8 Exponential Exponential Probability Distribution Probability Distribution Used when you are interested in! the interval in between occurrences. ! ! Continuous Probability Distributions - 7 Distribution Function: f (x) = ! e" ! x Usually focused on time (or a function of time) The reciprocal of the Poisson distribution Exponential random variables can be used to describe such things as: ! Time between computer crashes ! Time between vehicle arrivals at a toll booth ! Distance between potholes on the highway Where: Continuous Probability Distributions - 9 1 ! x/µ e µ (same as Poisson µ) Cumulative Probabilities: P(x ! x0 ) = 1" e" # x P(x > x0 ) = e! " x BUS210: Business Statistics Continuous Probability Distributions - 10 Exponential Exponential Probability Distribution Example: Al would like to know the probability that the time between two successive arrivals will be 2 minutes or less. f (x) = " = mean arrival rate Probability Distribution The time between arrivals of cars at Al s full-service gas pump follows an exponential probability distribution with a mean arrival of 1 car every 3 minutes. Note: Sometimes shown as e = 2.71828 Note: In this case, distance is a function of time (distance = speed x time) BUS210: Business Statistics for x # 0 f(x) P(x ! x0 ) = 1" e" # x = 1" 2.718 "2/3 = 1" .5134 = .4866 .4 .3 .2 .1 x 1 2 3 4 5 6 7 8 9 10 Time Between Successive Arrivals (mins.) BUS210: Business Statistics BUS210 Business Statistics Continuous Probability Distributions - 11 BUS210: Business Statistics Continuous Probability Distributions - 12 NSCC Chapter 7b 7b-3 Exponential Summary Probability Distribution Example: Customer Waiting Time ! Between 2 p.m. and 4 p.m. on Wednesday, patient insurance inquiries arrive at Blue Choice insurance at a mean rate of 2.2 calls per minute. What is the probability of waiting more than 30 seconds (i.e., 0.50 minutes) for the next call? Presented normal approximation of the binomial distribution ! Presented logarithmic probability distribution ! Applied normal approximation and the logarithmic distribution to problems Set " = 2.2 events/min and x = 0.50 min P(x > 0.50) = e–"x = e–(2.2)(0.5) = .3329 There is approximately a 33.3% chance of waiting more than 30 seconds for the next call. BUS210: Business Statistics BUS210 Business Statistics Continuous Probability Distributions - 13 BUS210: Business Statistics Continuous Probability Distributions - 14 NSCC